On the first evening I arrived at Cambridge University, almost three decades ago, I came out of my room to find two third-year men, in blazers and flannels, walking along the corridor ahead of me. I half-expected to overhear some fragment of impressive intellectual conversation. What I got was this: "Met a chap from a ... comprehensive today." There was a pause, before the speaker continued, in tones of some bafflement: "Seemed quite ... intelligent."

As I followed the men down the stairs to the dining hall I wanted to laugh. I thought I was witnessing the end of blind privilege, and the start of a new era. I had been in the first year of a comprehensive intake to a boys' grammar school, part of the wave of comprehensive education that had spread across England. There I'd witnessed the reluctant surrender of the old guard to a social and sexual mix. Now I was part of the first intake of women, and of comprehensive-educated men, in a men's college.

The college was suspiciously full of the offspring of its graduates, and of dull teenagers who had been expertly shovelled through the admissions process by their private schools. But it seemed obvious that the place would have to change. I felt the public-school stranglehold on Oxbridge would inevitably be weakened by the intellectual competition from the newly confident state schools. I could not have been more wrong. Last week's analysis of Oxbridge entry by the Sutton Trust showed that just 100 schools, four-fifths of them private, provide one-third of all Oxbridge undergraduates. Only two of the schools on the list are comprehensives, and even they are the kinds of comprehensives that manage to screen their entry.

At the top of the league, schools such as Westminster and St Paul's are sending half their sixth forms to Oxbridge every year. Overall, the state/private split in the intake has scarcely shifted from a state-school intake of 56% at Oxford in 1980 and 50% at Cambridge in 1979, to the figure close to 54% that it is at Oxbridge today. Despite decades of announcements from governments that they intend to widen educational opportunity, bright children from poor or ordinary backgrounds are still finding it very hard to get through.

The trust argues that something is clearly going wrong with the admissions process, because its statistics show that the differential rates of admission cannot be justified by A-level grades alone. On the whole, private schools are securing a far higher proportion of places for their straight-A students than state schools are. The trust says that this really matters, because Oxbridge graduates still have a distinct advantage in the jobs market. It's simply unjust that those advantages should go so disproportionately to a small elite. The question is: why is it happening, and what, if anything, can be done about it?

The trust believes that the reasons for the discrepancy lie in a combination of bias, low expectations, the exclusion of poor children from selective schools, and the lack of preparation for Oxbridge entry. It has made remarkable efforts to combat all four. But its critique of the system doesn't go far enough.

State-school children will never get to elite universities in the numbers that they should as long as state schools are forced to teach in the way that they do. What the top universities are looking for, besides academic performance, is intellectual creativity, a capacity for lateral thought and argument, and a deep knowledge of and enthusiasm for the subject. Some private schools have the time, resources, and the carefully selected intake, which enable them to provide that; many others are expert in training their pupils in the skills they need to fake it. They make it a priority because this is what their market, and their customers, demand. In contrast, the state system is answerable to government, and its priority has become the delivery of the test results and statistics which prove that education is a success. In state schools, what most children learn is that as long as they memorise what they are told for tests, and repeat the key words on the mark schemes in exams, then a questioning approach and wider reading are neither necessary nor welcome. The breadth, depth and articulacy which the best universities seek is not being taught.

It is a rare teacher and a rare school that, faced with large classes, children of all needs and abilities, and a very prescriptive, bureaucratic curriculum, have the time and the capacity to give their pupils more than the system requires. More typical is the experience of a history teacher, in an apparently excellent state school, who finished teaching his 14-year-olds about the first world war on a Tuesday. The following Thursday the class began studying the rise of Nazi Germany, 1933-39. After 20 minutes, one child put her hand up to ask what had happened between 1918 and 1933. "We really don't have the time to go into that now," the teacher said. So they never did.

The limitations of this kind of approach are increasingly being acknowledged by the government's own agencies. This summer Ofsted issued a report on the teaching of history. It said that a "successful curriculum" had been "faithfully delivered". And what was the result of this success? Why, in Ofsted's own words, that young people "could not answer the big questions of history", that they had "little sense of how events connected", that their knowledge was "patchy", their "sense of chronology weak", and that "they are generally unable to reflect on themes and issues, or relate a longer story of the history of Britain, Europe or elsewhere over an extended period of time".

This style of teaching may not be quite so devastating for middle-class children, who already inhabit a culture outside school which expects them to talk, argue and read. It is disastrous for the children from poorer homes, the very group that educational policy has been trying so hard to reach. Without a context or a wider purpose to their learning, and without the chance to learn how to discuss ideas and issues, these children are extremely unlikely to come through their school years with the capacity to compete for university places against the articulate and the carefully prepared. The statistics show that, on the whole, they don't.

Some people will want to dismiss the Sutton Trust report as a minor irrelevance; an argument about who gets to join a tiny elite. Others will believe that the sooner Oxford and Cambridge stop being fetishised as the most desirable universities to go to, the better. Both arguments have some merit. But the report's statistics illustrate the wider and more important point, which matters to all of us: that our schools aren't focusing on what really matters in education - the developing of every child's curiosity and talents.

Yesterday in Bournemouth, Gordon Brown put the unlocking of the nation's talents at the core of his speech. If he's really serious, that will require radical change: more money, far smaller classes, and a new focus on what children learn rather than what they are taught. But change is vital, because 10 years after Labour took power, it's still true that class is destiny in Britain.